Finite Elements in Electromagnetics

3. Eddy currents and skin effect

Oszkár Bíró

IGTE, TU Graz

Kopernikusgasse 24Graz, Austria

email: biro@igte.tu-graz.ac.at

OverviewEddy current problemsFormulations in eddy current free regionsFormulations in eddy current regionsCoupling of formulationsSkin effect problemsVoltage excitation, A,V-A formulationCurrent excitation, T,- formulation

n: nonconducting region

Air

J(r,t) = 0

Coil

J(r,t) known

0

c: eddy current regionJ(r,t) unknown

Typical eddy current problem

Maxwell’s equations:0JH curl ,

0Bdiv i n n ,

JH curl ,BE jcurl ,

0Jdiv ,0Bdiv i n

c .

BHHB ,i n

n a n d i n c ,

EJ , JE i n c .

A s s u m p t i o n : 00 JT curl .

Boundary conditions

n : o u t e r b o u n d a r y o f n

c : o u t e r b o u n d a r y o f

c

0nH o r 0 nB o n n ,

0nH o r 0nE o n c .

o rnTnH 0 o r nTnB 0 o n

n ,nTnH 0 o r 0nE o n

c .

Interface conditions

nc: interface betw een n and c

nH and nB are continuous on nc.

Magnetic scalar potential in n

grad0TH ,

en

kkkt

1

NT 0

D i f f e r e n t i a l e q u a t i o n :)()( 0T divgraddiv i n

n .

B o u n d a r y c o n d i t i o n s :0 o r nTn 0 grad o n

n .

Finite element approximation

nn

kkk

n N1

)(

G a l e r k i n e q u a t i o n s :

n n

dgradNdgradgradN in

i 0T )( ,

i = 1 , 2 , . . . , n n

Magnetic vector potential in n

AB curl .

D i f f e r e n t i a l e q u a t i o n

0TA curlcurlcurl )( i n n

B o u n d a r y c o n d i t i o n s :0nA o r nTnA 0 curl o n

n .

Finite element approximation

en

kkk

n a1

)( NAA

G a l e r k i n e q u a t i o n s :

n n

dcurldcurlcurl in

i 0TNAN )( ,

i = 1 , 2 , . . . , n e

P o s i t i v e s e m i d e f i n i t e m a t r i x

Magnetic vector potential alone in c

*AB curl , *AE j .

D i f f e r e n t i a l e q u a t i o n0AA ** jcurlcurl i n

c .

B o u n d a r y c o n d i t i o n s :0nA * o r nTnA 0 *curl o n

c .

Finite element approximation

en

kkk

n a1

)(** NAA

G a l e r k i n e q u a t i o n s :

c c

djdcurlcurl ni

ni

)(*)(* ANAN

c

dcurl i 0TN , i = 1 , 2 , . . . , n e

N o n s i n g u l a r b u t i l l - c o n d i t i o n e d m a t r i x

Magnetic vector and electric scalar potential in c

AB curl , gradVjj AE .

D i f f e r e n t i a l e q u a t i o n s :

0AA gradVjjcurlcurl ,

0)( gradVjjdiv A i n c .

B o u n d a r y c o n d i t i o n s :

0nA o r nTnA 0 curl ,

0VV = c o n s t a n t o r 0)( gradVjj An

o n c .

Finite element approximation

en

kkk

n a1

)( NAA ,

nn

kkk

n NVVV1

)( .

G a l e r k i n e q u a t i o n s :

c c

djdcurlcurl ni

ni

)()( ANAN

cc

dcurldgradVj in

i 0TNN )( ,

i = 1 , 2 , . . . , n e ,

c

dgradNj ni

)(A

0)( c

dgradVgradNj ni , i = 1 , 2 , . . . , n n

S i n g u l a r s y s t e m b u t i m p r o v e d c o n d i t i o n i n g .

Current vector and magnetic scalar potential in c

gradTTH 0 , TTJ 0 curlcurl .

D i f f e r e n t i a l e q u a t i o n s :

gradjjcurlcurl TT

00 TT jcurlcurl ,

)()( 0TT divjgraddivj i n c .

B o u n d a r y c o n d i t i o n s :

0nT o r nTnT 0 curlcurl ,

0 = c o n s t a n t o r 0)( TnTn grad

o n c .

Finite element approximation

en

kkk

n t1

)( NTT ,

nn

kkk

n N1

)( .

G a l e r k i n e q u a t i o n s :

c c

djdcurlcurl ni

ni

)()( TNTN

c

dgradj ni

)(N

cc

djdcurlcurl ii 00 TNTN ,

i = 1 , 2 , . . . , n e ,

c

dgradNj ni

)(T

c

dgradgradNj ni

)(

c

dgradNj i 0T , i = 1 , 2 , . . . , n n

S i n g u l a r s y s t e m b u t g o o d c o n d i t i o n i n g .

Coupling A,V in c to A in n: A,V-A formulation

I n t e r f a c e c o n d i t i o n s o n n c :

C o n t i n u i t y o f An nB i s c o n t i n u o u s

C o n t in u i t y o f nA curl i s a n a tu r a l i n t e r f a c ec o n d i t i o n nH i s c o n t i n u o u s

G a le r k in e q u a t i o n s r e m a in u n c h a n g e d

Coupling T, in c to in n: T,- formulation

In terface conditions on nc:

C ontinuity of and 0nT nH iscontinuous

C ontinuity of nT )( grad is a naturalin terface condition nB is continuous

G alerkin equations rem ain unchanged

Typical skin effect problem

u (t )

i ( t )

i ( t ) J ( r , t )

B ( r , t ) n : = 0

c : > 0

E 1 : 0nE

E 2 : 0nE

c n : nBnH , c o n t .

n

n

n HBEJ ,

,t

curl B

E

c : ,JH curl

n : ,JH curl

,0Bdiv

HB

Integral quantities, network parameters

2

),()(E

dtti nrJ

1

),(E

dt nrJ

c

dt

tpv

2),(

)(rJ

)()( 2 titR

dHdBtWnc

tB

m

),(

0

)(r

t

diLdd

i0

)()()(

dttdW

tptitutp mv

)()()()()(

Voltage excitation (1)

AB curl in nc and , gradUt

A

E in c

EH curl in c , 0H curl in n ,

)(, tuU 0nA on 1E , 0, U0nA on 2E ,

0nA or 0nH on )( nc

nA and nH are continuous on cn 0 nJ

Voltage excitation (2)

c

dt

tp

2),(

)(rJ

dHdBdtd

nc

tB ),(

0

r

c

dtp JE)(

nc

dt

HB

nc

dt

HB

nc

dt

curl HA

nc

dcurlt

HA

0

)(

nc

dt

nHA

c

dt

JA

Voltage excitation (3)

c

dt

tp JA

E)(

c

dgradU J

c

dgradUtp J)(

cnEEc

dUdUdiv21

0

nJJ

)()()()(1

titudtutpE

nJ

Boundary value problem for A,V (1)

Differential equations:

0A

A

tV

gradt

curlcurl 1

tV

U

0)(

tV

gradt

div A

0A

curlcurl1

in c,

in n,

Boundary value problem for A,V (2)Boundary conditions:

t

dutV0

)()(, 0nA on 1E ,

0)(, tV0nA on 2E ,

0nA or 0nA curl1

on )( nc .

Interface conditions:

nA and nAcurl1

are continuous on cn .

Current excitation (1)

in c

in c ,

in n .

on )( nc

and are continuous and cn

,TTJ 0 curl gradTTH 0,

grad0TH

0B

J

tcurl

1 0Bdiv in n ,

)0( nB0nE or

nTnH 0

nB 0nT on

,

.

Properties of T0

0T0 curl nin ,

1E

dcurl nT0 2E

dcurl nT0 )(ti

nT0 is continuous on .cn

A possible choice of T0

Solve the static current field in c

i ( t ) 00 TJ curl

n : = 0 c : > 0 C

C

n

ti3

)(

4)(

)()(:Q

QQS0

rr

rrdsrHrT

nHnT S0 :cn

0T 0

curlcurlc

1:

0nT 0 curlE 1

:1

0nT 0 curlE 1

:2

Boundary value problem for T, (1)

Differential equations:

in c,

in n,

0TTTT 00

grad

tcurlcurl

1

0)(

graddivt

TT0

0)(

graddivt 0T

Boundary value problem for T, (2)Boundary conditions:

2E

or on )( nc .

Interface conditions:

and

on 1E ,

are continuous on cn .

0,1 nTT0nT 0 gradcurl

0nT on cn .

0 0nT0 grad

nT0 grad